Simplicial complexes of whisker type

Let $I\subset K[x_1,\dots ,x_n]$ be a zero-dimensional monomial ideal, and $\mathrm{\Delta }(I)$ be the simplicial complex whose Stanley–Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\mathrm{\Delta }(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $\mathrm{\Delta }(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley–Reisner ideal of the Alexander dual of $\mathrm{\Delta }(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}L(I{)}^k$ and show that $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}L(I{)}^k=n$ for all $k\ge n$.